Question: Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{-5a^3 - 35a^2 + 40a}{2a^2 + 20a + 32}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ k = \dfrac {-5a(a^2 + 7a - 8)} {2(a^2 + 10a + 16)} $ $ k = -\dfrac{5a}{2} \cdot \dfrac{a^2 + 7a - 8}{a^2 + 10a + 16} $ Next factor the numerator and denominator. $ k = - \dfrac{5a}{2} \cdot \dfrac{(a + 8)(a - 1)}{(a + 8)(a + 2)}$ Assuming $a \neq -8$ , we can cancel the $a + 8$ $ k = - \dfrac{5a}{2} \cdot \dfrac{a - 1}{a + 2}$ Therefore: $ k = \dfrac{ -5a(a - 1)}{ 2(a + 2)}$, $a \neq -8$